Geometry of String Theory and Its Nonperturbative Effect

这三年的研究重点是弦场论对D-膜的解释，第一年(2001)，我从边界共形场理论的观点出发，考察了开弦星积的投影算符与D-膜的关系。在其他投影仪中，我们研究了身份、银色和蝴蝶操作符，ANC认为身份操作符似乎是最有前途的，因为它正确地再现了弦的张力。然而，同时也意识到，开放弦的中点的奇点阻止了严格的治疗。在第二年(2002)。在非对易几何中广泛使用的简化星积MoYAL积的基础上，我开始发展开放弦场论的正则化版本。在这种语言中，我发现奇点表现为星积的结合性的异常。我们不能以…的身份从异常中逃脱只要我们处理无限数量的分量，我们就引入了一个正则化方案，将分量域截断到有限个数。有了这个公式，我们可以可靠地计算所有的离壳层振幅，同时，我们还可以对快子真空进行解析研究。由于我们可以用有限秩矩阵将所有的振幅写成显式形式，因此脱壳振幅的计算是非常令人满意的。同时，我们认识到中点的奇异性实际上对快子真空的计算起主导作用。在这个意义上，我们意识到开放字符串语言不适合描述D-品牌。最后，我们开始分析封闭字符串语言中的D-品牌。众所周知，D-膜最好的描述是存在于封闭弦扇区的边界态。我们证明了所有已知的边界态都满足一个泛幂等关系，它与VSFT运动方程具有完全相同的形式。较少

The focus of the research in these three years is the interpretation of D-brane in terms of the string field theory.In the first year (2001), I started the research by examining the relation between the projection operators of the open string star product and D-brane in the viewpoint of the boundary conformal field theory. Among other projectors, we examined identity, sliver and butterfly operators anc argued that the identity operator seems the most promising one since it reproduces the string tension correctly. At the same time, however, realized that a singularity at the midpoint of the open string prevents the rigorous treatment.In the second year (2002). I began to develop a regularized version of the open string field theory based on the simplified star product, Moyal product, which is used extensively in the noncommutative geometry. In this language, I found that the singularity appears as an anomaly of the Associativity of the star product. We can not escape from the anomaly as … More long as we work with the infinite number of components and we introduced a regularization scheme by cutting off the component field to the finite number. With this reformulation, we can make a reliable computation of all the off-shell amplitudes: At the same time, we can also make an analytic study of the tachyon vacuum. The computation of off-shell amplitude is quite satisfactory since we can write all the amplitudes in the explicit form with finite rank matrices. At the same time, we realized that the singularity from the midpoint is actually dominant to the computation of the tachyon vacuum. In this sense, we realized that open string language is not appropriate to describe the D-brane.Finally, we start to analyze the D-brane in the closed string language. It is known that D-brane is best described by the boundary state which live in the closed string sector. We show that all the known boundary state satisfy a universal idempotency relation, which has exactly the same form of the VSFT equation of motion. Less